Qualitative Analysis of a Tumor-Immune System with Antigen Delay and Michaelis-Menten Type Inhibition Term

In this paper, we discuss a class of dynamic models of the interaction between tumors and the immune system with antigen delay and Michaelis-Menten type inhibition terms. The Michaelis-Menten type function $\frac{\beta E\left(t\right)T\left(t\right)}{a+T\left(t\right)}$ and $\frac{\alpha E\left(t\right)T\left(t\right)}{a+T\left(t\right)}$ is used to describe the immune response of effector cells interacting with tumor cells, the linear antigen stimulation term $cT$ is used to describe the linear recruitment effect of tumor antigens on effector cells, and the stimulation delay of tumor antigens in the immune system is introduced. The dynamic behavior of the model is studied through qualitative analysis and numerical simulation. Saddle-node bifurcation may occur both in the case with and without time delay. Contrary to the case without time delay, stimulation delay may lead to some complex dynamic behaviors and biological phenomena. In the presence of time delay, the existence condition of Hopf bifurcation at the equilibrium point is obtained. Further discussion shows that the model may exhibit bistability under certain conditions, that is, the growth and development state of the tumor depends on its initial state. Finally, numerical simulation is used to verify the accuracy of the relevant theoretical results, and the corresponding biological significance is briefly discussed.